Math Analysis Transformation of Functions

In this lesson, our instructor Vincent Selhorst-Jones teaches about the Transformation of Functions. This lesson teaches vertical and horizontal shift, stretch, shrink, and flip. Youll get a summary of transformations and learn about how order matters when stacking transformations. The lesson ends with four examples for additional practice.

If you horizontally stretch a function does that mean you vertically stretch the functions graph and visa versa. for example the inverse function 1/x transformed to 1/2x could this be interpreted as multiplying f(x) by 1/2 ( vertical shrink) and also interpretted as f(2x) horizontal shrink. is this just coincidence ?

its just visually if i stretch a graph horizontally shouldn't the y values decrease as in a vertical shrink and if I shrink horizontally shouldnt the y value increase.

1 answer

Last reply by: Professor Selhorst-JonesMon Oct 28, 2013 9:59 AM

Post by Charles Reinmuthon October 27, 2013

The vertical stretch and horizontal stretch look very similar to me. I see a difference in that there are parentheticals around the horizontal (eg. f(x) = (3x)^2 ...vs... f(x) = 3x^2)

Still, I don't think I understand fully what is going on. What exactly is the difference? Perhaps I missed something. Thankyou so much!!

1 answer

Last reply by: Professor Selhorst-JonesSun Jul 28, 2013 9:11 PM

Post by Jason Toddon July 26, 2013

Professor, in example 2 how did you differentiate vertical vs. horizontal flip possibilities? Thanks in advance.

1 answer

Last reply by: Professor Selhorst-JonesThu Jul 11, 2013 1:11 PM

Post by Sarawut Chaiyadechon June 28, 2013

Thanks

1 answer

Last reply by: Professor Selhorst-JonesThu May 23, 2013 10:47 AM

Post by Matthew Chantryon May 22, 2013

These questions are for Example 2:

1. Shouldn't everything in the h(x) function after the - be in brackets?2. Could this be seen as a horizontal flip as well? Would that look different?

Thank-you

Transformation of Functions

We often have to work with functions that are similar to ones we already know, but not precisely the same. Many times, this difference is the result of a transformation. A transformation is a shift, stretch, or flip of a function.

A vertical shift moves a function up or down by some amount. If we want to shift a function f by k units, we use

f(x) + k.

[If k is positive, it moves up. If negative, down.]

A vertical stretch/shrink "pulls/pushes" the function away from/toward the x-axis by some multiplicative factor. If we want to vertically stretch/shrink a function by a multiplicative factor a, we use

a ·f(x).

[If a > 1, the function stretches. If 0 < a < 1, it shrinks. If a=1, nothing happens.]

A horizontal shift moves a function left or right by some amount. If we want to shift a function f by k units, we use

f(x+k).

[If k is positive, the graph moves left. If k is negative, the graph moves right. (This may seem counter-intuitive, but remember that the shift is being caused by how f "sees" (x+k). Check out the video for an in-depth explanation of what's going on.)]

A horizontal stretch/shrink changes how fast the function "sees" the x−axis. If we want to horizontally stretch/shrink a function by a multiplicative factor a, we use

f(a ·x).

[If a > 1, it shrinks horizontally ("speeds up"). If 0 < a < 1, it stretches horizontally ("slows down"). (This may seem counter-intuitive, but remember that the stretch/shrink is being caused by how f "sees" (a·x). Check out the video for an in-depth explanation of what's going on.)]

To vertically flip a graph (mirror over the x-axis), we need to swap every output for the negative version. If we want to vertically flip, we use

−f(x).

To horizontally flip a graph (mirror over the y-axis), we need to "flip" how f "sees" the x-axis. We do this by plugging in −x (which is effectively a "flipped" x). If we want to horizontally flip a function, we use

f(−x).

If you want to do multiple transformations, just apply one transformation after another. However, order matters, so start by deciding on the order you want the transformations to occur in. Then apply them to the base function in that order.

Transformation of Functions

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.